Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 53, pp. 1–18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu TRAVELING WAVE SOLUTIONS FOR FULLY PARABOLIC KELLER-SEGEL CHEMOTAXIS SYSTEMS WITH A LOGISTIC SOURCE RACHIDI B. SALAKO, WENXIAN SHEN Abstract. This article concerns traveling wave solutions of the fully parabolic Keller-Segel chemotaxis system with logistic source, ut = ∆u− χ∇ · (u∇v) + u(a− bu), x ∈ RN , τvt = ∆v − λv + µu, x ∈ RN , where χ, µ, λ, a, b are positive numbers, and τ ≥ 0. Among others, it is proved that if b > 2χµ and τ ≥ 1 2 (1 − λ a )+, then for every c ≥ 2 √ a, this system has a traveling wave solution (u, v)(t, x) = (Uτ,c(x · ξ − ct), V τ,c(x · ξ − ct)) (for all ξ ∈ RN ) connecting the two constant steady states (0, 0) and (a b , µ λ a b ), and there is no such solutions with speed c less than 2 √ a, which improves the results established in [30], and shows that this system has a minimal wave speed c∗0 = 2 √ a, which is independent of the chemotaxis. 1. Introduction This work concerns traveling wave solutions of the fully parabolic chemotaxis system ut = ∆u− χ∇ · (u∇v) + u(a− bu), x ∈ RN , τvt = ∆v − λv + µu, x ∈ RN , (1.1) where χ, µ, λ, a, b are positive real numbers, τ is a nonnegative real number, and u(t, x) and v(t, x) denote the concentration functions of some mobile species and chemical substance, respectively. Biologically, the positive constant χ measures the sensitivity effect on the mobile species by the chemical substance which is produced overtime by the mobile species. The reaction term u(a− bu) in the first equation of (1.1) describes the local dynamics of the mobile species. λ represents the degradation rate of the chemical substance. µ is the rate at which the mobile species produces the chemical substance. The constant 1/τ in the case τ > 0 measures the diffusion rate of the chemical substance, and the case τ = 0 is supposed to model the situation when the chemical substance diffuses very quickly. System (1.1) is a simplified version of the chemotaxis system proposed by Keller and Segel in [18, 19]. Chemotaxis models describe the oriented movements of bi- ological cells and organisms in response to certain chemical substances. These 2010 Mathematics Subject Classification. 35B35, 35B40, 35K57, 35Q92, 92C17. Key words and phrases. Parabolic chemotaxis system; logistic source; traveling wave solution; minimal wave speed. c©2020 Texas State University. Submitted August 11, 2019. Published May 27, 2020. 1 2 R. B. SALAKO, W. SHEN EJDE-2020/53 mathematical models play very important roles in a wide range of biological phe- nomena and accordingly a considerable literature is concerned with their mathe- matical analysis. The reader is referred to [11, 12] for some detailed introduction into the mathematics of Keller-Segel models. One of the central problems about (1.1) is whether a positive solution blows up at a finite time. This problem has been studied in many papers in the case that a = b = 0 (see [11, 14, 16, 17, 25, 38, 39, 40]). It is known that finite time blow-up may occur if either N = 2 and the total initial population mass is large enough, or N ≥ 3. It is also known that some radial solutions to (1.1) in plane collapse into a persistent Dirac-type singularity in the sense that a globally defined measure-valued solution exists which has a singular part beyond some finite time and asymptotically approaches a Dirac measure (see [23, 34]). We refer the reader to [2, 13] and the references therein for more insights in the studies of chemotaxis models. When the constants a and b are positive, the finite time blow-up phenomena in (1.1) may be suppressed to some extent. In fact in this case, it is known that when the space dimension is equal to one or two, solutions to (1.1) on bounded domains with Neumann boundary conditions and initial functions in a space of certain integrable functions are defined for all time. And it is enough for the self limitation coefficient b to be large enough compared to the chemotaxis sensitivity coefficient to prevent finite time blow-up, see [15, 31, 35]. Traveling wave solutions constitute another class of important solutions of (1.1). Observe that, when χ = 0, the first equation in chemotaxis system (1.1) reduces to ut = ∆u+ u(a− bu), x ∈ RN . (1.2) Due to the pioneering works of Fisher [7] and Kolmogorov, Petrowsky, Piskunov [20] on traveling wave solutions and take-over properties of (1.2), (1.2) is also referred to as the Fisher-KPP equation. The following results are well known about traveling wave solutions of (1.2). Equation (1.2) has traveling wave solutions of the form u(t, x) = φ(x · ξ − ct) (ξ ∈ SN−1) connecting 0 and a b (φ(−∞) = a b , φ(∞) = 0) of all speeds c ≥ 2 √ a and has no such traveling wave solutions of slower speed. c∗0 = 2 √ a is therefore the minimal wave speed of traveling wave solutions of (1.2) connecting 0 and a b . Since the pioneering works by Fisher [7] and Kolmogorov, Petrowsky, Piscunov [20], a huge amount of research has been carried out toward the front propagation dynamics of reaction diffusion equations of the form ut = ∆u+ uf(t, x, u), x ∈ RN , (1.3) where f(t, x, u) < 0 for u� 1, ∂uf(t, x, u) < 0 for u ≥ 0; see [1, 3, 4, 5, 6, 8, 9, 21, 22, 24, 26, 27, 32, 33, 36, 37, 41]. In [30], the authors of the current paper studied the existence of traveling wave solutions of (1.1) connecting the two constant steady states (0, 0) and (ab , µ λ a b ). Roughly, in [30], it is proved that when the chemotaxis sensitivity χ is small relative to the logistic damping b, (1.1) has traveling wave solutions connecting (0, 0) and (ab , µ λ a b ) with speed c, which is bounded below by some constant c∗ > c∗0 = 2 √ a and is bounded above by some constant c∗∗ < ∞. But many fundamental questions remain open, for example, whether (1.1) has traveling wave solutions connecting (0, 0) and (ab , µ λ a b ) with speed c � 1; whether there is a minimal wave speed of traveling wave solutions of (1.1) connecting (0, 0) and (ab , µ λ a b ), and if so, how the chemotaxis affects the minimal wave speed. EJDE-2020/53 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 3 The objective of this article is to investigate those fundamental open questions. To state the main results of the current paper, we first introduce the definition of traveling wave solutions of (1.1) and the induced problems to be studied. 1.1. Traveling wave solutions and induced problems. An entire solution of (1.1) is a classical solution (u(t, x), v(t, x)) of (1.1) which is defined for all x ∈ RN and t ∈ RN . Note that the constant solutions (u(t, x), v(t, x)) = (0, 0) and (u(t, x), v(t, x)) = (ab , µa λb ) are clearly two particular entire solutions of (1.1). An entire solution of (1.1) of the form (u(t, x), v(t, x)) = (Uτ,c(x ·ξ−ct), V τ,c(x ·ξ−ct)) for some unit vector ξ ∈ SN−1 and some constant c ∈ R is called a traveling wave solution with speed c. A traveling wave solution (u(t, x), v(t, x)) = (Uτ,c(x · ξ − ct), V τ,c(x · ξ − ct)) (ξ ∈ SN−1) of (1.1) with speed c is said to connect (0, 0) and (ab , µa λb ) if lim inf x→−∞ Uτ,c(x) = a b and lim sup x→∞ Uτ,c(x) = 0. (1.4) We say that a traveling wave solution (u(t, x), v(t, x)) = (Uτ,c(x · ξ − ct), V τ,c(x · ξ − ct)) of (1.1) is nontrivial and connects (0, 0) at one end if lim inf x→−∞ Uτ,c(x) > 0 and lim sup x→∞ Uτ,c(x) = 0. (1.5) Observe that for given c ∈ R, a traveling wave solution (u(t, x), v(t, x)) = (Uτ,c(x· ξ − ct), V τ,c(x · ξ − ct)) (ξ ∈ SN−1) of (1.1) with speed c connecting the states (0, 0) and (ab , µa λb ) gives rise to a stationary solution (u, v) = (Uτ,c(x), V τ,c(x)) of the parabolic-elliptic system ut = uxx + ((c− χvx)u)x + (a− bu)u, x ∈ R, 0 = vxx + τcvx − λv + µu, x ∈ R. (1.6) connecting the states (0, 0) and (ab , µa λb ). Conversely, if (u, v) = (Uτ,c(x), V τ,c(x)) is a stationary solution of (1.6) connect- ing the states (0, 0) and (ab , µa λb ), then (u(t, x), v(t, x)) = (Uτ,c(x · ξ− ct), V τ,c(x · ξ− ct)) is a traveling wave solution of (1.1) with speed c connecting the states (0, 0) and (ab , µa λb ) for any ξ ∈ SN−1. To study traveling wave solutions of (1.1) with speed c connecting the states (0, 0) and (ab , µa λb ) is then equivalent to study stationary solutions of (1.6) connecting the states (0, 0) and (ab , µa λb ). It is clear that (1.6) is equivalent to ut = uxx + (c− χvx)ux + (a− χvxx − bu)u, x ∈ R, 0 = vxx + τcvx − λv + µu, x ∈ R. (1.7) Hence, to study traveling wave solutions of (1.1) connecting the states (0, 0) and (ab , µa λb ) we shall study steady state solutions of (1.7) connecting the states (0, 0) and (ab , µa λb ). Before stating the main results of the current paper, we next recall some existing results on the existence of solutions of (1.7) with given initial functions and existence of steady state solutions of (1.7) or traveling wave solutions of (1.1) connecting the states (0, 0) and (ab , µa λb ). 1.2. Existing results. Let Cbunif(R) = { u ∈ C(R) : u(x) is uniformly cont. in x ∈ R and sup x∈R |u(x)| <∞ } equipped with the norm ‖u‖∞ = supx∈R |u(x)|. 4 R. B. SALAKO, W. SHEN EJDE-2020/53 Proposition 1.1 (Local solution). For every nonnegative initial function u0 in Cbunif(R) and c in R, there is a unique maximal time Tmax(u0), such that (1.7) has a unique classical solution (u(t, x;u0, c), v(t, x;u0, c)) defined for every x ∈ R and 0 ≤ t < Tmax(u0) with u(0, x;u0, c) = u0(x). Moreover if Tmax(u0) <∞ then lim t→Tmax(u0)− ‖u(t, ·;u0, c)‖∞ =∞. The above proposition can proved by similar arguments as those in [31, Theorem 1.1]. The following proposition follows from the arguments of [30, Theorems A and B] (it is proved in [30, Theorems A and B] for the case that λ = µ = 1). Proposition 1.2 (Global solution). Consider (1.7). (1) Assume that 0 ≤ χµτc 2 √ λ < b− χµ. Then for any u0 ∈ Cbunif(R) with 0 ≤ u0, Tmax(u0) =∞. Moreover, ‖u(t, ·;u0, c)‖∞ ≤ max{‖u0‖∞, a b− χµ− χµτc 2 √ λ } for every t ≥ 0. (2) Assume that 0 ≤ χµτc√ λ < b − 2χµ. Then for any u0 ∈ Cbunif(R) with infx∈R u0(x) > 0, lim t→∞ [ ‖u(t, ·;u0, c)− a b ‖∞ + ‖v(·, t;u0, c)− µ λ a b ‖∞ ] = 0. Proposition 1.3. (1) For every τ > 0, there is 0 < χ∗τ < b 2µ such that for every 0 < χ < χ∗τ , there exist two positive numbers 0 < c∗(χ, τ) < c∗∗(χ, τ) satisfying that for every c ∈ (c∗(χ, τ) , c∗∗(χ, τ)), (1.1) has a traveling wave solution (u, v) = (U(x · ξ − ct), V (x · ξ − ct)) (∀ξ ∈ SN−1) connecting the constant solutions (0, 0) and (ab , µ λ a b ). Moreover, lim χ→0+ c∗∗(χ, τ) =∞, lim χ→0+ c∗(χ, τ) = { 2 √ a if 0 < a ≤ λ+τa (1−τ)+ λ+τa (1−τ)+ + a(1−τ)+ λ+τa if a ≥ λ+τa (1−τ)+ , lim x→∞ U(x; τ) e−κx = 1, where κ is the only solution of the equation κ + a κ = c in the interval (0,min{ √ a, √ λ+τa (1−τ)+ }). (2) For any given τ ≥ 0 and χ ≥ 0, (1.1) has no traveling wave solutions (u, v) = (U(x · ξ− ct), V (x · ξ− ct)) (∀x ∈ SN−1) with (U(−∞), V (−∞)) = (ab , µ λ a b ), (U(∞), V (∞)) = (0, 0), and c < 2 √ a. As mentioned before, in the absence of chemotaxis (i.e. χ = 0), c∗0 = 2 √ a is the minimal wave speed of the Fisher-KPP equation (1.2). Both biologically and mathematically, it is interesting to know whether the results stated in Proposition 1.3(1) can be improved to the following: for any c ≥ c∗0, (1.1) has a traveling wave solution (u(t, x), v(t, x)) = (U(x ·ξ−ct), V (x ·ξ−ct)) (for all ξ ∈ SN−1) connecting (ab , µ λ a b ) and (0, 0), which implies that (1.1) has a minimal wave speed, and the chemotaxis does not affect the magnitude of the minimal wave speed. Also, as mentioned before, this article is to investigate the above open problem or to improve the results obtained in [30]. Roughly, we will show that there is no EJDE-2020/53 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 5 upper bound for the speeds of traveling wave solutions of (1.1) and under some natural conditions, c∗0 = 2 √ a is the minimal wave speed of (1.1). The precise statements of the main results are stated in next subsection. 1.3. Statements of main results. To state our main results, we first introduce some notation. For given a c ∈ R, let Bλ,c,τ = 1√ 4λ+ τ2c2 , λc1 = (τc+ √ 4λ+ τ2c2) 2 , λc2 = ( √ 4λ+ τ2c2 − τc) 2 , cκ = a+ κ2 κ , ∀0 < κ < √ a. Note that λc2 and −λc1 are the positive and negative roots of the quadratic equation m2 + τcm− λ = 0. Note also that λc1λ c 2 = λ and λc1 + λc2 = 1 Bλ,c,τ . (1.8) All the above quantities are defined for any τ ≥ 0. Throughout this work, we suppose that c > 0. This restriction is justified by the fact that (1.1) does not have a non-trivial traveling wave with speed c ≤ 0 (see Proposition 1.3(2)). Note that, by (1.8), λc2Bλ,c,τ λc2 + κ ( κ− λ λc1 ) + = λc2(κ− λc2)+ (λc2 + λc1)(κ+ λc2) < 1. (1.9) Hence the following quantity is well defined b∗τ = sup{1 + λcκ2 (κ− λcκ2 )+ (λcκ2 + λcκ1 )(κ+ λcκ2 ) : 0 < κ < √ a}. (1.10) It is clear that b∗τ is defined for all τ ≥ 0, b∗τ ≤ 2 for all τ ≥ 0, and b∗0 = 1+ ( √ a− √ λ)+ 2( √ a+ √ λ) . For the sake of simplicity in the statements of our results, let us introduce the following standing hypotheses. (H1) b > χµ. (H2) b > b∗τχµ. (H3) b > 2χµ. (H4) τ ≥ 1 2 ( 1− λ a ) + . Observe that (H3) implies (H2), and (H2) implies (H1). The following results about the existence of a global bounded classical solutions and the stability of the positive constant equilibria of (1.7) will be of great use in our arguments. Theorem 1.4. For any τ ≥ 0 and c > 0, the following hold. (i) If (H1) holds, then for every u0 ∈ Cbunif(R), with u0 ≥ 0, (1.7) has a unique global classical solution (u(t, x;u0, c), v(t, x;u0, c)) on (0,∞)×R satisfying limt→0+ ‖u(0, ·;u0, c)− u0(·)‖∞ = 0. Moreover it holds that ‖u(t, ·;u0, c)‖∞ ≤ max { ‖u0‖∞, a b− χµ } , t ≥ 0. (1.11) 6 R. B. SALAKO, W. SHEN EJDE-2020/53 (ii) If (H3) holds, then for every u0 ∈ Cbunif(R), with infx∈R u0(x) > 0, we have that lim t→∞ ( ‖u(t, ·;u0, c)− a b ‖∞ + ‖v(t, ·;u0, c)− aµ bλ ‖∞ ) = 0. (1.12) When τ = 0, we recover [31, Theorems 1.5 & 1.8]. For τ > 0, Theorem 1.4 improves the results stated in Proposition 1.2. Observe that the function (0, √ a) 3 κ 7→ λcκ1 − κ is strictly decreasing. Hence the quantity κ∗τ := sup{0 < κ < √ a |λcκ1 − κ ≥ 0} (1.13) is well defined. It holds that λcκ1 − κ > 0 whenever 0 < κ < κ∗τ . Note also that κ∗τ = min {√ a, √ λ+ τa (1− τ)+ } . (1.14) Indeed, it holds that λ c√a 1 > √ a for every τ ≥ 1. On the other hand, for 0 ≤ τ < 1, if λcκ1 = κ for some 0 < κ ≤ √ a, then it holds that λ+ κτcκ − κ2 = 0 ⇔ λ+ τa = (1− τ)κ2 with κ = √ λ+ τa 1− τ . Hence (1.14) holds. Let c∗(τ) = κ∗τ + a κ∗τ . (1.15) Note that κ∗τ and c∗(τ) are defined for all τ ≥ 0, and κ∗0 = min{ √ λ, √ a}, c∗(0) = min{ √ λ, √ a}+ a min{ √ λ, √ a} . We have the following theorem on the existence of traveling wave solutions of (1.1). Theorem 1.5. For any τ ≥ 0, the following hold. (1) If (H2) holds, then for any c > c∗(τ), (1.1) has a nontrivial traveling wave solution (u, v)(t, x) = (U(x · ξ− cκt), V (x · ξ− cκt)) (∀ ξ ∈ SN−1) satisfying (1.5), where κ ∈ (0, κ∗τ ) is such that cκ = c. Furthermore, it holds that lim x→∞ U(x) e−κx = 1. (1.16) If in addition (H3) holds, then lim x→−∞ |U(x)− a b | = 0. (1.17) (2) If (H2) and (H4) hold, then κ∗τ = √ a and c∗(τ) = 2 √ a. Hence for any c > 2 √ a, the results in (1) hold true. (3) Suppose that (H3) holds. Then system (1.1) has a traveling wave solution (u, v)(t, x) = (Uτ,c(x · ξ − ct, V τ,c(x · ξ − ct)) (for all ξ ∈ SN−1) with speed c∗(τ) connecting (0, 0) and (ab , aµ bλ ). Remark 1.6. (1) Note that the conditions in Proposition 1.3 are χ < χ∗τ and b > 2χµ, which imply both (H2) and (H3). Hence the assumptions in Theorem 1.5(1) are weaker than those in Proposition 1.3 for the existence of traveling wave solutions. Note also that, by Theorem 1.5(1), the lower bound c∗(τ) for the wave EJDE-2020/53 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 7 speed is independent of χ, and the upper bound is∞. By the proof of [30, Theorem C], κ∗τ = min{ √ a, λ+τa (1−τ)+ } is an upper bound found for the decay rate of traveling wave solutions found in [30]. Hence c∗(χ, τ) ≥ cκ∗τ = c∗(τ), that is, the lower bound provided in Theorem 1.5 for the wave speed of traveling wave solutions of (1.1) is not larger than that provided in Proposition 1.3. Moreover, under the assumptions (H2) and (H4), c∗(τ) = 2 √ a < c∗(χ, τ). Therefore Theorem 1.5 improves considerably Proposition 1.3. (2) Recall that b∗0 = 1 + ( √ a− √ λ)+ 2(a+ √ λ) , κ∗0 = min{ √ a, √ λ}, and c∗(0) = κ∗0 + a κ∗0 . Hence Theorem 1.5 in the case τ = 0 recovers [28, Theorem 1.4]. (3) When λ ≥ a, c∗(τ) = c∗0 = 2 √ a for any τ ≥ 0. Hence if λ ≥ a and 0 < χµ < b 2 hold, by Theorem 1.5 for every τ ≥ 0 and c ≥ 2 √ a, (1.1) has a traveling wave solution (u, v)(t, x) = (Uτ,c, V τ,c)(x − ct) with speed c connecting (0, 0) and (ab , aµ bλ ). Whence, if λ ≥ a and 0 < χ < b 2µ , Theorem 1.5 implies that c∗0 = 2 √ a is the minimal wave speed of traveling wave solutions of (1.1) connecting (0, 0) and (ab , aµ bλ ), and that the chemotaxis does not affect the magnitude of the minimal wave speed of (1.1). Biologically, λ ≥ a means that the degradation rate λ of the chemical substance is greater than the intrinsic growth rate a of the mobile species, and 0 < χµ < b 2 indicates that the product of the chemotaxis sensitivity χ and the rate µ at which the mobile species produces the chemical substance is less than half of the logistic damping b. (4) When λ < a, c∗(τ) = c∗0 = 2 √ a for τ > 1 2 (1 − λ a ). Hence if λ < a and 0 < χµ < b 2 hold, by Theorem 1.5 for every τ > 1 2 (1 − λ a ) and c ≥ 2 √ a, (1.1) has a traveling wave solution (u, v)(t, x) = (Uτ,c, V τ,c)(x− ct) with speed c connecting (0, 0) and (ab , aµ bλ ). Thus in this case, Theorem 1.5 also implies that c∗0 = 2 √ a is the minimal wave speed of traveling wave solutions of (1.1) connecting (0, 0) and (ab , aµ bλ ), and that the chemotaxis does not affect the magnitude of the minimal wave speed of (1.1). Biologically, τ > 1 2 (1− λ a ) indicates that diffusion rate of the chemical substance is not big. (5) By Theorem 1.5 it holds that c∗(τ) = 2 √ a whenever τ ≥ 1 2 and (1.1) has a minimal wave speed, which is c∗(τ). When λ < a and 0 ≤ τ < 1 2 , it remains open whether (1.1) has a minimal wave speed, and if so, whether the minimal wave speed equals 2 √ a. It would be interesting to study the stability of the traveling wave solutions of (1.1). When τ = 0, the spreading speeds of solutions of (1.1) with compactly supported initial functions are studied in [28]. It would be also interesting to study these spreading results when τ > 0, which we plan to carry out in our future work. The rest of this article is organized as follows. In Section 2, we prove some preliminaries results to use in the subsequent sections. Section 3 is devoted to the proof of Theorem 1.4, while Section 4 is devoted to the proof of Theorem 1.5. 2. Preliminary lemmas In this section, we prove some lemmas to be used in the proofs of the main results in the later sections. Throughout of this section, we assume τ ≥ 0. For u ∈ Cbunif(R) and c ∈ R, let Ψ(x;u, c, τ) = µ ∫ ∞ 0 ∫ R e−λse− |x+τcs−y|2 4s √ 4πs u(y) dy ds. (2.1) 8 R. B. SALAKO, W. SHEN EJDE-2020/53 It is well known that Ψ(x;u, c, τ) ∈ C2 unif(R) and solves the elliptic equation d2 dx2 Ψ(x;u, c, τ) + τc d dx Ψ(x;u, c, τ)− λΨ(x;u, c, τ) + µu = 0. Lemma 2.1. It holds that Ψ(x;u, c, τ) = µ√ 4λ+ τ2c2 ∫ R e − √ 4λ+τ2c2|x−y|−τc(x−y) 2 u(y)dy =µBλ,c,τ ( e−λ c 1x ∫ x −∞ eλ c 1yu(y)dy + eλ c 2x ∫ ∞ x e−λ c 2yu(y)dy ) (2.2) and d dx Ψ(x;u, c, τ) = µBλ,c,τ ( − λc1e−λ c 1x ∫ x −∞ eλ c 1yu(y)dy + λc2e λc2x ∫ ∞ x e−λ c 2yu(y)dy ) . (2.3) Proof. For the case τ = 0, this lemma is proved in [28, Lemma 2.1]. In the following, we prove the case that τ > 0. Observe that it is sufficient to prove the result for τ = 1. The general case follows by replacing c by τc. So, without loss of generality, we set τ = 1. First, observe that the following identity holds,∫ ∞ 0 e− β2 4s −s √ 4πs ds = e−β 2 , ∀β > 0. (2.4) Next using Fubini’s Theorem, one can exchange the order of integration in (2.1) to obtain Ψ(x;u, c, 1) =µ ∫ ∞ 0 ∫ R e−λse− |x+cs−y|2 4s [4πs] 1 2 u(y) dy ds =µ ∫ R [ ∫ ∞ 0 e− |x+cs−y|2 4s −λs √ 4πs ds ] u(y)dy = ∫ R e− c(x−y) 2 [ ∫ ∞ 0 e− [ (x−y)2 4s + (4λ+c2) 4 s ] √ 4πs ds ] u(y)dy (2.5) By the change of variables z = (4λ+c2)s 4 and taking β = √ 4λ+c2 2 |x − y|, from (2.4) it follows that∫ ∞ 0 e− [ (x−y)2 4s + (4λ+c2) 4 s ] √ 4πs ds = 2√ 4λ+ c2 ∫ ∞ 0 e− β2 4z −z √ 4πz dz = 1√ 4λ+ c2 e− √ 4λ+c2|x−y| 2 . This together with (2.5) implies that Ψ(x;u, c, 1) = µ√ 4λ+ c2 ∫ R e − √ 4λ+c2|x−y|−c(x−y) 2 u(y) dy. Thus (2.2) holds. Note that (2.3) then follows from a direction calculation. � Lemma 2.2. For every u ∈ Cbunif(R), u(x) ≥ 0, it holds that | d dx Ψ(x;u, c, τ)| ≤ λc1Ψ(x;u, c, τ), ∀x ∈ R, c ∈ R. (2.6) EJDE-2020/53 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 9 Furthermore, it holds that χκΨx(·;u, c, τ)− χΨxx(·;u, c, τ) ≤ Mχµ eκx (Bλ,c,τ ((τc+ κ)λ2 − λ)+ (λ2 + κ) + 1 ) (2.7) whenever 0 ≤ u(x) ≤Me−κx for some κ ≥ 0 and M > 0. In particular, if χµ (Bλ,c,τ((τc+ κ)λ2 − λ ) + (λ2 + κ) + 1 ) ≤ b, (2.8) then χκΨx(x;u, c, τ)− χΨxx(x;u, c, τ)− bMe−κx ≤ 0, ∀x ∈ R, (2.9) whenever 0 ≤ u(x) ≤Me−κx for some positive real numbers κ > 0 and M > 0. Proof. For the case that τ = 0, this lemma is proved in [28, Lemma 2.2]. In the following, we prove the lemma for any τ ≥ 0. First, by (2.2) and (2.3), we have | d dx Ψ(x;u, c, τ)| ≤ √ 4λ+ τ2c2 + τc 2 Ψ(x;u, c, τ). This implies (2.6). Next, we prove (2.9). It follows from (2.1) and (2.3) that χκΨx(x;u, c, τ)− χΨxx(x;u, c, τ) (2.10) = χκΨx(x;u, c, τ)− χ(λΨ(x;u, c, τ)− τcΨx(x;u, c, τ)− µu) (2.11) = χ(τc+ κ)Ψx(x;u, c, τ)− χλΨ(x;u, c, τ) + χµu (2.12) = −χµBλ,c,τ ((τc+ κ)λc1 + λ) e−λ c 1x ∫ x −∞ eλ c 1yu(y)dy (2.13) + χµBλ,c,τ ((τc+ κ)λc2 − λ) eλ c 2x ∫ ∞ x e−λ c 2yu(y)dy + χµu. (2.14) Hence, since 0 ≤ u ≤Me−κx, it follows that χ (κΨx(x;u, c, τ)−Ψxx(x;u, c, τ)) ≤ χµBλ,c,τ ((τc+ κ)λc2 − λ)+Meλ c 2x ∫ ∞ x e−λ c 2ye−κydy + χµM eκx = χµ (Bλ,c,τ ((τc+ κ)λc2 − λ)+ (λc2 + κ) + 1 ) Me−κx Hence, (2.7) follows. � Remark 2.3. Observe that τcλc2 − λ = τc 2 (√ 4λ+ τ2c2 − τc ) − λ = 2λτc√ 4λ+ τ2c2 + τc − λ =− λλc2 λc1 < 0. (2.15) 10 R. B. SALAKO, W. SHEN EJDE-2020/53 Hence Bλ,c,τ λc2 (τcλc2 − λ)+ = 0, Bλ,c,τ λc2 + κ ( (τc+ κ)λc2 − λ ) + = λc2Bλ,c,τ λc2 + κ ( κ− λ λc1 ) + . We also note from (1.8) that Bλ,c,τ ( λ λc1 + λ λc2 ) = 1. (2.16) These identities will be frequently used later. For 0 < κ < κ̃ < √ a with κ̃ < 2κ and M,D ≥ 1, consider the functions ϕκ(x), Uκ,D(x), and Uκ,D(x) given by ϕκ(x) = e−κx, (2.17) U−D (x) = ϕκ(x)−Dϕκ̃(x), x ∈ R, (2.18) Uκ,M (x) = min{M,ϕκ(x)}, (2.19) Uκ,D(x) = { ϕκ(x)−Dϕκ̃(x), x ≥ xκ,D ϕκ(xκ,D)−Dϕκ̃(xκ,D), x ≤ xκ,D, (2.20) where xκ,D satisfies max{ϕκ(x)−Dϕκ̃(x) : x ∈ R} = ϕκ(xκ,D)−Dϕκ̃(xκ,D). (2.21) Letting xκ,D := ln(D) κ̃−κ , it holds that U−D (x) { > 0 if x > xκ,D, < 0, if x < xκ,D. For u ∈ Cbunif(R), let Au,c(U) = Uxx + (c− χΨx(·;u, c, τ))Ux + (a− χΨxx(·;u, c, τ)− bU)U. (2.22) Lemma 2.4. For a given τ ≥ 0, assume that (H2) holds and 0 < κ < κ∗τ . Then there is D∗ > 1 such that for every D ≥ D∗, M > 0, and u ∈ Ẽ := {u ∈ Cbunif(R) : max{U−D (x), 0} ≤ u(x) ≤ min{M,ϕκ(x)};∀x ∈ R} it holds that Au,cκ(U−D ) ≥ 0 ∀x ∈ (xκ,D,∞). (2.23) Proof. We first note that (H2) implies (2.8), and κ < κ∗τ implies λcκ1 > κ. (2.24) Let u ∈ Ẽ be given and U−(x) = U−D (x). Then Au,cκ(U−) = U−xx + (cκ − χΨx(·;u, cκ))U−x + (a− χΨxx − bU−)U− = ( κ2e−κx − κ̃2De−κ̃x ) + (cκ − χΨx)(−κe−κx + κ̃De−κ̃x) + a(e−κx −De−κ̃x) − (χΨxx + bU−)U− = D(κ̃cκ − κ̃2 − a) ek̃x − χΨx(κ̃De−κ̃x − κe−κx)− (χ(λΨ− µu− τcκΨx) + bU−)U− EJDE-2020/53 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 11 = DAκe −k̃x − χΨx(−κe−κx + κ̃De−κ̃x)− (χλΨ− χµu− τcκχΨx + bU−)U− ≥ DAκe−k̃x + χΨx(κe−κx − κ̃De−κ̃x)︸ ︷︷ ︸ I1 + (−χλΨ + τcκχΨx − (b− χµ)U−)U−︸ ︷︷ ︸ I2 . where Aκ := κ̃cκ − κ̃2 − a. Next, observe that since λcκ1 > κ, it folows that I1 =µBλ,cκ,τ ( λcκ2 e−λ cκ 2 x ∫ ∞ x u(y) eλ cκ 2 x − λcκ1 eλ cκ 1 x ∫ x −∞ u(y) e−λ cκ 1 y dy ) (κe−κx − κ̃De−κ̃x) ≥− µBλ,cκ,τ ( κλcκ1 e(λ cκ 1 +κ)x ∫ x −∞ eλ cκ 1 yu(y)dy + κ̃Dλcκ2 e−(λ cκ 2 −κ̃)x ∫ ∞ x e−λ cκ 2 yu(y) ) ≥− µBλ,cκ,τ ( κλcκ1 e(λ cκ 1 +κ)x ∫ x −∞ eλ cκ 1 ye−κydy + κ̃Dλcκ2 e−(λ cκ 2 −κ̃)x ∫ ∞ x e−λ cκ 2 ye−κy ) =− µBλ,cκ,τ ( κλcκ1 λcκ1 − κ e−(2κ−κ̃)x + κ̃Dλcκ2 λcκ2 + κ e−κx ) e−κ̃x and I2 + (b− χµ)(e−2κx −De−(κ̃+κ)x) χµBλ,cκ,τ = ( (τcκ − λ)λcκ2 e−λ cκ 2 x ∫ ∞ x u(y) eλ cκ 2 y dy − (τcκ + λ)λcκ1 eλ cκ 1 x ∫ x −∞ u(y) e−λ cκ 1 y dy ) U− + (b− χµ)D χµBλ,cκ,τ U−(x)e−κ̃x ≥− ( (τcκ + λ)λcκ1 e(λ cκ 1 +κ)x ∫ x −∞ u(y) e−λ cκ 1 y dy + (τcκ − λ)−λ cκ 2 U −(x) e−λ cκ 2 x ∫ ∞ x u(y) eλ cκ 2 y dy ) ≥− ( (τcκ + λ)λcκ1 e(λ cκ 1 +κ)x ∫ x −∞ u(y) e−λ cκ 1 y dy + (τcκ − λ)−λ cκ 2 e−(λ cκ 2 −κ)x ∫ ∞ x u(y) eλ cκ 2 y dy ) ≥− ( (τcκ + λ)λcκ1 e(λ cκ 1 +κ)x ∫ x −∞ e(λ cκ 1 −κ)ydy + (τcκ − λ)−λ cκ 2 e−(λ cκ 2 −κ)x ∫ ∞ x e−(λ cκ 2 +κ)ydy ) =− ( (τcκ + λ)λcκ1 λcκ1 − κ + (τcκ − λ)−λ cκ 2 λcκ2 + κ ) e−2κx. Thus, with D > 1, 0 < κ1 := 2κ− κ̃ < κ, and x > xκ,D > 0, it holds that A(U−) e−κ̃x ≥DAκ − [ χµBλ,c,τ ( (κ+(τcκ+λ))λ cκ 1 λcκ1 −κ + (κ̃D+(τcκ−λ)+)λcκ2 λcκ2 +κ ) + (b− χµ) ] eκ1xκ,D . Setting κ̃ = κ+ η, we have Aκ > 0 and e−κ1xκ,D = e− (κ−η) η ln(D) = 1 D κ−η η . Therefore, for 0 < η < min{κ2 , √ a− κ}, it holds that κ < κ̃ = κ+ η < min{2κ, √ a}, κ− η η > 1, lim D→∞ ( DAκ − [ χµBλ,c,τ ( (κ+D(c+λ))λcκ1 λcκ1 −κ + (κ̃D+(cκ−λ)+)λcκ2 λcκ2 +κ ) + (b− χµ) ] eκ1xκ,D ) =∞. 12 R. B. SALAKO, W. SHEN EJDE-2020/53 Therefore, there is D∗ > 1 such that (2.23) holds for every D ≥ D∗ and u ∈ Ẽ . � 3. Proof of Theorem 1.4 Proof of Theorem 1.4. (1) Let (u(t, x;u0, c), v(t, x;u0, c)) be defined on [0, Tmax). Note by Proposition 1.1 that to show that Tmax =∞, it is sufficient the prove that (1.11) holds. For every T ∈ (0, Tmax) let MT := sup0≤t≤T ‖u(t, ·;u0, c)‖∞. With κ = 0 and M = MT , it follows from (2.7) that ut ≤ uxx + (c− χvx)ux + ( a+ χµ (Bλ,c,τ (τcλc2 − λ)+ λc2 + 1 ) MT − bu ) u, for 0 < t < T . Hence, by the comparison principle for parabolic equations, it holds that ‖u(t, ·;u0, c)‖∞ ≤ max { ‖u0‖∞, a+ χµ (Bλ,c,τ (τcλc2−λ)+ λc2 + 1 ) MT b } , ∀t ∈ [0, T ]. Hence, if MT > ‖u0‖∞, we must have MT ≤ a+ χµ (Bλ,c,τ (τcλc2−λ)+ λc2 + 1 ) MT b . By (2.15), (τcλc2 − λ)+ = 0. Hence MT ≤ a b− χµ . Therefore, MT ≤ max { ‖u0‖∞, a b− χµ } , 0 < T < Tmax, which yield that Tmax =∞, and by Remark 2.3 we conclude that (1.11) holds. (2) We show that (1.12) holds. We follow the ideas of the proof of [31, Theorem 1.8]. Let u = lim sup t→∞ ‖u(t, ·;u0, c)‖∞ and u := lim inf t→∞ inf x∈R u(t, x;u0, c). Since infx∈R u0(x) > 0, it follows from the arguments in [29, Theorem 1.2 (i) ] that 0 < u ≤ u <∞. It suffices to prove that u = ū = a b . (3.1) To this end, for T > 0, let uT := sup t≥T sup x∈R u(t, x;u0, c) and uT := inf t≥T inf x∈R u(t, x;u0, c). Let L(u) = uxx + (c− χvx)ux. By (2.10) (with κ = 0), for every t ≥ T and x ∈ R, it holds ut − L(u) + (b− χµ)u2 ≤ ( a− χµBλ,c,τ (τcλc1 + λ)e−λ c 1x ∫ x −∞ eλ c 1yuT dy ) u + χµBλ,c,τ ( (τcλc2 − λ)+ e−λ c 2x ∫ ∞ x uT eλ c 2y dy − (τcλc2 − λ)− e−λ c 2x ∫ ∞ x uT eλ c 2y dy ) u = ( a+ χµBλ,c,τ ( (τc− λ λc2 )+uT − (τc+ λ λc1 )uT − (τc− λ λc2 )−uT )) u. EJDE-2020/53 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 13 Hence, by comparison principle for parabolic equations, it holds that (b− χµ)u ≤ a+ χµBλ,c,τ ( (τc− λ λc2 )+uT − (τc+ λ λc1 )uT − (τc− λ λc2 )−uT ) . Letting T →∞, we obtain (b− χµ)u ≤ a+ χµBλ,c,τ ( (τcλc2 − λ)+ λc2 u− (τcλc1 + λ) λc1 u− (τcλc2 − λ)− λc2 u ) . (3.2) Similarly, from (2.10) (with κ = 0) it follows for every t ≥ T and x ∈ R that ut − L(u) + (b− χµ)u2 ≥ ( a− χµBλ,c,τ (τcλc1 + λ)e−λ c 1x ∫ x −∞ eλ c 1yuT dy ) u + χµBλ,c,τ ( (τcλc2 − λ)+ e−λ c 2x ∫ ∞ x uT eλ c 2y dy − (τcλc2 − λ)− e−λ c 2x ∫ ∞ x uT eλ c 2y dy ) u = ( a+ χµBλ,c,τ ( (τc− λ λc2 )+uT − (τc+ λ λc1 )uT − (τc− λ λc2 )−uT )) u. Hence, by the comparison principle for parabolic equations, it folows that (b− χµ)u ≥ a+ χµBλ,c,τ ( (τc− λ λc2 )+uT − (τc+ λ λc1 )uT − (τc− λ λc2 )−uT ) . Letting T →∞, we obtain that (b−χµ)u ≥ a+χµBλ,c,τ ( − (τcλc1 + λ) λc1 u+ (τcλc2 − λ)+ λc2 u− (τcλc2 − λ)− λc2 u ) . (3.3) Since (τcλc2 − λ)+ = 0 by (2.15), by adding side-by-side inequalities (2.10) and (3.2), we obtain (b− χµ)(u− u) ≤χµBλ,c,τ (τcλc1 + λ λc1 + (λ− τcλc2) λc2 ) (u− u) =χµBλ,c,τ ( λ λc1 + λ λc2 ) (u− u). By (2.16), we have Bλ,c,τ ( λ λc1 + λ λc2 ) = 1. Thus, since (H3) holds, we conclude that u = u. By (2.15), (3.2), and (3.3), we have (b− χµ)u = a+ χµBλ,c,τ ( − (τcλc1 + λ) λc1 u+ τcλc2 − λ λc2 u ) = a− χµu. This implies (3.1), and (2) thus follows. � 4. Proof of Theorem 1.5 In this section, following the techniques developed in [30], we present the proof of Theorem 1.5. Without loss of generality, we assume that N = 1 in (1.1). Through this section we suppose that (H2) holds and 0 < κ < κ∗τ . We choose 0 < η < min{2κ, √ a − κ} and set κ̃ = κ + η and M = a b−χµ . We fix a constant D ≥ D∗, where D∗ is given by Lemma 2.20. Define E := {u ∈ Cbunif(R) : Uκ,D ≤ u ≤ Uκ,M} 14 R. B. SALAKO, W. SHEN EJDE-2020/53 where Uκ,M and Uκ,D are given by (2.19) and (2.23) respectively. For u ∈ E , we let U(t, x;u) denote the solution of the parabolic equation Ut = Au,cκ(U), x ∈ R, t > 0 U(0, x) = Uκ,M , x ∈ R. (4.1) Lemma 4.1. (i) For every u ∈ Ẽ, the function U(t, x) ≡ M satisfies the inequality Au,cκ(U) ≤ 0 on R× R. (ii) For every u ∈ Ẽ, the function U(t, x) = e−κx satisfies Au,cκ(U) ≤ 0 on R× R. (iii) For every u ∈ Ẽ, the function U(t, x) = U−D , where U−D is given by (2.18), satisfies Au,cκ(U) ≥ 0 on R× (xκ,D,∞). (iv) Suppose that (H3) holds. There 0 < δ � 1 such that for every u ∈ Ẽ, the function U(t, x) = δ satisfies Au,cκ(U) ≥ 0 on R× R. The proof of the above lemma follows from Lemmas 2.2 and 2.4. Proof of Theorem 1.5. (1) Thanks to Lemma 4.1, for D � D∗, it follows by the comparison principle for parabolic equations that U(t2, x;u) < U(t1, x;u), ∀x ∈ R, 0 ≤ t1 < t2, ∀u ∈ Ẽ . Hence the function U(x;u) = lim t→∞ U(t, x;u, cκ), u ∈ Ẽ is well defined. Moreover, by estimates for parabolic equations, it follows that Uxx + (cκ −Ψx(·;u, cκ))Ux + (a− χΨxx(·;u, cκ)− bU)U = 0, x ∈ R, and U(·;u, cκ) ∈ Ẽ ∀u ∈ Ẽ . Next we endow Ẽ with the compact open topology. From this point, it follows from the arguments of the proof of [30, Theorem 4.1] that the function Ẽ 3 u 7→ U(·;u, cκ) ∈ Ẽ is compact and continuous. Hence, by the Schauder’s fixed point theorem, it has a fixed point, say u∗. Clearly, (u, v)(t, x) = (u∗,Ψ(·;u∗, cκ))(x − cκt) is a nontrivial traveling wave solution of (1.1) satisfying (1.16). The proof that lim inf x→−∞ u∗(x) > 0 follows from [10, Theorem 1.1 (i)]. If (H3) holds, it follows from Lemma 4.1 (iv) that for D � D∗, it holds that E 3 u 7→ U(·;u, cκ) ∈ E . Hence lim inf x→−∞ u∗(x) > 0. Therefore, by the stability of the positive constant equilibrium established in The- orem 1.4, it follows that lim x→−∞ u∗(x) = a b . This completes the proof of Theorem 1.5 (1). EJDE-2020/53 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 15 (2) Observe that c∗(τ) = cκ∗τ , and, by (1.14), κ∗τ = min {√ a, √ λ+ τa (1− τ)+ } . This implies that, if λ ≥ a or τ ≥ 1, κ∗τ = √ a and then c∗(τ) = 2 √ a. In the case λ < a and τ < 1, (H4) implies that τ ≥ 1 2 ( 1 − λ a ) . This implies that 2τa ≥ a − λ and then a ≤ λ+τa 1−τ . Hence we also have κ∗τ = √ a and c∗(τ) = 2 √ a. (2) then follows from (1). (3) Let {cn}n≥1 be a sequence of real numbers satisfying cn > c∗(τ) and cn → c∗(τ) as n → ∞. For each n ≥ 1, let (U cn,τ (x), V cn,τ (x)) denote a traveling wave solution of (1.1) with speed cn connecting (0, 0) and (ab , aµ bλ ) given by Theorem 1.5 (1). For each n ≥ 1, since the set {x ∈ R : U cn,τ (x) = a 2b} is bounded and closed, hence compact, then it has a minimal element, say xn. Next, consider the sequence {Un(x), V n(x)}n≥1 defined by (Un(x), V n(x)) = (U cn,τ (x+ xn), V cn,τ (x+ xn)), ∀x ∈ R, n ≥ 1. Then, for every n ≥ 1, (u(t, x), v(t, x)) = (Un(x− cnt), V n(x− cnt)) it a traveling wave solution of (1.1) with speed cn satisfying Un(−∞) = a b , Un(∞) = 0, Un(0) = a 2b , and Un(x) ≥ a 2b for every x ≤ 0. Note that ‖Un‖∞ = ‖U cn,τ‖∞ ≤ a b− χµ , ∀n ≥ 1. Hence by estimates for parabolic equations, without loss of generality, we may suppose that (Un, V n) → (U∗, V ∗) locally uniformly in C2(R). Moreover, the function (U∗, V ∗) satisfies 0 = U∗xx + (c∗(τ)− χV ∗x )U∗x + (a− χV ∗xx − bU∗)U∗, x ∈ R 0 = V ∗xx + τc∗(τ)V ∗x − λV ∗ + µU∗, x ∈ R, (4.2) and ‖U∗‖∞ ≤ a b− χµ , U∗(0) = a 2b , U∗(x) ≥ a 2b ∀x ≤ 0, U∗(x) > 0, ∀x ∈ R. (4.3) Hence, since (H3) holds, it follows by the stability of the positive constant equilib- rium giving by Theorem 1.4 (2) that lim x→−∞ U∗(x) = a b . So, to complete this proof, it remains to show that lim sup x→∞ U∗(x) = 0. (4.4) Suppose by contradiction that (4.4) does not hold. Whence, there is a sequence {yn}n≥1 with y1 = 0, yn < yn+1, yn →∞ as n→∞, and lim n→∞ U∗(yn) = lim sup x→∞ U∗(x) > 0. (4.5) Consider a sequence {zn}n≥1 given by U∗(zn) = min{U∗(z) | yn ≤ z ≤ yn+1}, ∀n ≥ 1. 16 R. B. SALAKO, W. SHEN EJDE-2020/53 Thus lim n→∞ U∗(zn) = inf x∈R U∗(x). Note that infx∈R U ∗(x) = 0, otherwise since (H3) holds, we would have from Theo- rem 1.4 (2) that U∗(x) ≡ a b , which contradicts to (4.3). Thus, there is some n0 � 1 such that zn is a local minimum point for every n ≥ n0, and hence U∗xx(zn) ≥ 0 and U∗x(zn) = 0, ∀n ≥ n0. (4.6) By (4.3), ‖U∗‖∞ ≤ a b−χµ , then it follows from the first equation of (4.2), from (2.10) with κ = 0 and M = a b−χµ , that 0 ≥ U∗xx + (c∗ − χV ∗x )U∗x + ( a− χµBλ,c∗,τ ( λ λc ∗ 1 + λ λc ∗ 2 ) a b− χµ − (b− χµ)U∗ ) U∗, where c∗ = c∗(τ), which combined with (2.16) yield 0 ≥ U∗xx + (c∗(τ)− χV ∗x )U∗x + (a(b− 2χµ) b− χµ − (b− χµ)U∗ ) U∗. 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Zlatoš; Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl. (9) 98 (2012), no. 1, 89-102. Rachidi B. Salako Department of Mathematics, The Ohio State University, Columbus, OH 43210-1174, USA Email address: salako.7@osu.edu Wenxian Shen Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA Email address: wenxish@auburn.edu 1. Introduction 1.1. Traveling wave solutions and induced problems 1.2. Existing results 1.3. Statements of main results 2. Preliminary lemmas 3. Proof of Theorem 1.4 4. Proof of Theorem 1.5 Acknowledgments References